Metric (度量) $d: X^2 \to \mathbb{R}_{\ge 0}$ on a topological space $(X, \mathcal{T})$ is a non-negative bivariate function that satisfies:
Psudo-metric is a mapping satisfying all the requirements of a metric except non-degeneracy. For example, for the space $C(\mathbb{R})$ of continuous real functions, $\rho_n(f, g) = \sup_{x \in [-n,n]} |f(x) - g(x)|$, $n \in \mathbb{N}$, are psudo-metrics. But they can be transformed into a metric, e.g. $\sigma(f, g) = \sum_{n=1}^{\infty} 2^{-n} \min \{1,\rho_n(f, g) \}$.
Metric space (度量空间) $(X, d)$ is a set $X$ with a metric $d: X \times X \to \mathbb{R}_{\ge 0}$. Metric specifies the distance among elements within a space. Subspace $(A, d)$ of a metric space $(X, d)$, where $A \subset X$, is also a metric space. Product space $(X \times Y, d_x \times d_y)$ of two metric spaces $(X, d_x)$ and $(Y, d_y)$, where $d_x \times d_y ((x_1, y_1), (x_2, y_2)) = d_x(x_1, x_2) + d_y(y_1, y_2)$, is also a metric space.
Distance between a point and a set in a metric space $(X, d)$ is the mapping $d: X \times \mathcal{P}(X) \to \mathbb{R}{\ge 0}$ such that $d(x, A) = \inf{y \in A} d(x,y)$. Diameter of a set in a metric space $(X, d)$ is the mapping $\mathrm{diam}: \mathcal{P}(X) \to \mathbb{R}_{\ge 0}$ such that $\mathrm{diam}(A) = \sup_{x,y\in A} d(x,y)$.
Bounded space is a metric space whose diameter is finite. Totally-bounded space is a metric space that can be represented as a finite union of arbitrarily bounded subspaces: $\forall \varepsilon > 0$, $\exists \{x_i\}_{i=1}^{n} \subset X$: $X \subset \cup_{i=1}^n B_\varepsilon(X_i)$.
Lp metric is the metric induced from an Lp norm.
Euclidean metric...
Isometry (等距同构) or congruent transformation is a distance-preserving bijective map between two metric spaces: given metric spaces $(X, d_x)$ and $(Y, d_y)$, bijection $f: X \to Y$ is an isometry iff $d_x(x_1, x_2) = d_y(f(x_1), f(x_2)), \forall x_1, x_2 \in X$. For example, bending a plane is an isometry. Two metric spaces are isometric if and only if there exists an isometry between them.
Convergent sequence is a sequence $\{x_n\}_{n \in \mathbb{N}}$ in a metric space $(X, d)$ such that $\exists x \in X$, $\forall \varepsilon > 0$, $\exists N \in \mathbb{N}:$ $\forall n > N$, $d(x_n, x) < \varepsilon$; and we say the sequence converges to $x$. A sequence in a set may converge in one metric but not in another.
Uniform convergence.
Cauchy sequence is a sequence $\{a_i\}_{i=1}^N$ in a metric space that satisfies: $\forall \varepsilon > 0$, $\exists N \in \mathbb{N}:$, $\forall m, n > N$, $d(a_m, a_n) < \varepsilon$. Every convergent sequence in a metric space is a Cauchy sequence. Complete metric space is a metric space where every Cauchy sequence converges. The real line and the complex plane are complete metric spaces, but the rational numbers is not. Completeness of the real line is the main reason why it is used in calculus instead of smaller sets e.g. the rational line. A subspace of a complete metric space is complete if and only if it is a closed subset.
Completion of metric space: Every metric space is isometric to a dense subspace of a unique complete metric space up to isometries. Completion $(\hat X, \hat d)$ of a metric space $(X, d)$ is the complete metric space with a dense subspace isometric to the metric space. The completion of the rational numbers is the real line: $\hat{\mathbb{Q}} = \mathbb{R}$.
Theorem: (equivalence of metric and topological definitions of complete space).
Baire's Theorem.
Contraction mapping theorem.
Completion.
Equivalent sequence.
Theorem: (Completion of metric space).
Open ball $B_r(x)$ of radius $r$ centered at $x$ in a metric space $(X, d)$ is the set of points with distance to $x$ less than $r$: $B_r(x) = \{y \in X \mid d(y, x) < r\}$, $r > 0$. Closed ball $B_r\left[x\right]$ is the set of points with distance to $x$ no greater than $r$: $B_r\left[x\right] = \{y \in X \mid d(y, x) \le r\}$, $r \ge 0$. Sphere $S_r\left[x\right]$ is the set of points with distance to $x$ equal $r$: $S_r\left[x\right] = \{y \in X \mid d(y, x) = r\}$, $r \ge 0$.
Topology generated by a metric or metric topology $\mathcal{T}_d$ is the topology $\mathcal{T(B)}$ generated by the class $\mathcal{B}$ of open balls in a metric space $(X, d)$: $\mathcal{T}_d = \mathcal{T(B)}$, $\mathcal{B} = \{B_r(x) \mid x \in X, r > 0\}$. Euclidean topology, usual topology, or ordinary topology on $\mathbb{R}^n$ is the topology generated by the Euclidean metric in $\mathbb{R}^n$.
Sequentially compact.
Boltzano-Weierstras property
Covering, sub-covering, open covering, compact.
Covering number $N(S,s)$ of a metric space $S$ with disks at radius $s$ is the minimal number of such disks needed to cover the space. This number is finite if the metric space is compact.
Compactness Theorem: equivalence of sequentially compact, compact, Boltzano-Weierstras property, H-compact
Theorem: continuous mapping from a compact space induces another compact space.
Theorem: product space of compact spaces is compact.
Boltzano-Weierstras theorem
Theorem: real continuous mapping from a compact space has maximum and minimum.
Pointwise compace, equi-continuous.
Theorem: (Arzela-Ascoli)
In case of function spaces, compact spaces typically are similar to finite-dimensional space.
Theorem: (equivalence of metric and topological definitions of continuous mapping)
continuity, uniform continuity.